[Term Entry] Add documentation for PyTorch .sinh() function #8139

content/pytorch/concepts/tensor-operations/terms/sinh/sinh.md

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+---
+Title: '.sinh()'
+Description: 'Computes the hyperbolic sine of each element in a PyTorch tensor.'
+Subjects:
+  - 'Machine Learning'
+  - 'Data Science'
+Tags:
+  - 'PyTorch'
+  - 'Deep Learning'
+  - 'Machine Learning'
+  - 'Tensors'
+CatalogContent:
+  - 'intro-to-py-torch-and-neural-networks'
+  - 'paths/computer-science'
+---
+
+The **`.sinh()`** function in PyTorch computes the hyperbolic sine of each element in the input tensor. This operation applies the mathematical function $\sinh(x) = \frac{e^x - e^{-x}}{2}$ element-wise to all values in the tensor.
+
+The hyperbolic sine function is commonly used in neural network activation functions, numerical methods, and scientific computing applications involving hyperbolic trigonometric transformations.
+
+## Syntax
+
+```pseudo
+torch.sinh(input, *, out=None) → Tensor
+```
+
+**Parameters:**
+
+- `input`: The input tensor containing elements for which the hyperbolic sine will be computed.
+- `out` (optional): A tensor to store the output. If provided, the result is written to this tensor.
+
+**Return value:**
+
+Returns a new tensor where each element is the hyperbolic sine of the corresponding element in the input tensor.
+
+## Example 1: Single Element Tensor
+
+This example demonstrates computing the hyperbolic sine of a single-element tensor:
+
+```py
+import torch
+
+# Create a single-element tensor
+x = torch.tensor([2.0])
+
+# Compute the hyperbolic sine
+result = torch.sinh(x)
+
+print("Input:", x)
+print("sinh(2.0):", result)
+```
+
+This produces the following output:
+
+```
+Input: tensor([2.])
+sinh(2.0): tensor([3.6269])
+```
+
+## Example 2: 1D Array
+
+This example shows how to compute the hyperbolic sine of a 1D tensor:
+
+```py
+import torch
+
+# Create a 1D tensor with various values
+x = torch.tensor([0.0, 1.0, -1.0, 2.0, -2.0])
+
+# Compute the hyperbolic sine
+result = torch.sinh(x)
+
+print("Input tensor:", x)
+print("Hyperbolic sine:", result)
+```
+
+This produces the following output:
+
+```
+Input tensor: tensor([ 0.,  1., -1.,  2., -2.])
+Hyperbolic sine: tensor([ 0.0000,  1.1752, -1.1752,  3.6269, -3.6269])
+```
+
+Note that the hyperbolic sine function is antisymmetric, meaning $\sinh(-x) = -\sinh(x)$, which is why `sinh(1.0)` and `sinh(-1.0)` have opposite signs but equal magnitudes.
+
+## Example 3: Multi-Dimensional Array
+
+This example demonstrates computing the hyperbolic sine of a 2D tensor:
+
+```py
+import torch
+
+# Create a 2D tensor (2x3 matrix)
+x = torch.tensor([[0.0, 0.5, 1.0],
+                  [1.5, 2.0, 2.5]])
+
+# Compute the hyperbolic sine
+result = torch.sinh(x)
+
+print("Input tensor:")
+print(x)
+print("\nHyperbolic sine:")
+print(result)
+```
+
+This produces the following output:
+
+```
+Input tensor:
+tensor([[0.0000, 0.5000, 1.0000],
+        [1.5000, 2.0000, 2.5000]])
+
+Hyperbolic sine:
+tensor([[ 0.0000,  0.5211,  1.1752],
+        [ 2.1293,  3.6269,  6.0502]])
+```
+
+The `.sinh()` function preserves the shape of the input tensor, applying the hyperbolic sine operation element-wise to each value in the multi-dimensional array.